This paper investigates the role of the function x|x| as a chaos generator in nonautonomous systems. A Duffing-like nonautonomous oscillator is used for illustration. It is rigorously proven via the Melnikov function method that this particular quadratic function induces Smale horseshoes to the Duffing-like system. Moreover, its physical meaning as an energy function is demonstrated, which provides a critical value for the emergence of chaos. Simulations with bifurcation analysis are given for better understanding of the underlying dynamics.
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Man et al. (2001) studied this question.
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